Mathematicians Solve Decades-Old Puzzle About Network Phase Transitions

In the week before Christmas 2025, five mathematicians at ETH Zurich -- Sahar Diskin, Philip Easo, Ritvik Ramanan Radhakrishnan, Benny Sudakov and Vincent Tassion -- completed a proof of one of the biggest open problems in percolation theory, the study of flow in a network. By the morning of December 17 the group was convinced its argument was correct, and the full proof was finished by Christmas, answering a decades-old question about how fast a percolation network floods once opened up.

Percolation models rest on a device built by Simon Broadbent and John Hammersley to study carbon filters in gas masks. Take a lattice, a grid of evenly spaced points, and flip a coin for every pair of neighboring points: heads connects them with an edge fluid can flow through, tails blocks it. How far fluid travels depends on the probability of heads. Below a threshold called the critical probability, fluid collects in small, isolated puddles; above it, the lattice opens up -- a phase transition, like liquid water turning to ice.

The field traces back to coal: in the 1940s, Rosalind Franklin, later known for her work on the structure of DNA, studied coal's tiny holes at the British Coal Utilization Research Association. Researchers later proposed the "sharpness conjecture" -- that puddles stay tiny below the critical probability while a single ocean dominates almost immediately above it. Two independent groups, one in New Jersey and one in Moscow, proved sharpness on lattices in the 1980s, a result Tom Hutchcroft of Princeton University and the California Institute of Technology called "foundational."

In 1996, Itai Benjamini of the Weizmann Institute for Science and collaborator Oded Schramm asked whether sharpness also holds for a broader class of networks called transitive graphs. The new ETH Zurich proof addresses that long-open question across a huge variety of graphs. Asaf Nachmias of Tel Aviv University, who studies percolation theory and probability, called the result "stunning."